What is a tablet?

By a tablet (the name by analogy with tablature) is meant a pair [n, c] consisting of an approximate note-number n (an integer) and a chord denoting element c, typically a tuple of integers, which defines a type of chord up to octave equivalence. Together they define a note in a just intonation group or regular temperament. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a val or val-like mapping, the number n in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see Composing with tablets.

There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below.

The 5-limit 3et tablet

If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted |* e3 e5>, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that [r e3 e5] defines a major triad with root given by |* e3 e5> when r is even, and a minor triad when r is odd.

If r is even, therefore, we will regard the 3-tuple [r e3 e5] as denoting a major triad, and if [n, [r e3 e5]] is the tablet we wish to define, we first calculate u = n - 5e3 - 7e5. Then if r is even, we have

• If u mod 3 = 0, then

note(n, [r e3 e5]) = |u/3 e3 e5>

• If u mod 3 = 1, then

note(n, [r e3 e5]) = |(u-7)/3 e3 e5+1>

• If u mod 3 = 2, then

note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5>

On the other hand, if r is odd then

• If u mod 3 = 0, then

note(n, [r e3 e5]) = |u/3 e3 e3-e5>

• If u mod 3 = 1, then

note(n, [r e3 e5]) = |(u-7)/3+3 e3+1 e3-e5-1>

• If u mod 3 = 2, then

note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5>

Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying <3 5 7|note(n, c) = n.

4et tablets

The 7-limit 4et tablet

Suppose m0, m1, m2 and m3 are four monzos denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being |* e3 e5 e7>, where the asterisk can be any integer value, then m = |* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord |* e3 e5 e7> then m = |* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal. Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the cubic lattice of 7-limit tetrads.

If t = [n, [a b c]] is a tablet, define u = n - 7a - 4b - 2c. If a+b+c is even, then

• If u mod 4 = 0, then

note(t) = |u/4 (-a+b+c)/2 (a-b+c)/2 (a+b-c)/2>

• If u mod 4 = 1, then

note(t) = |(u-9)/4 (-a+b+c)/2 1+(a-b+c)/2 (a+b-c)/2>

• If u mod 4 = 2, then

note(t) = |(u-6)/4 1+(-a+b+c)/2 (a-b+c)/2 (a+b-c)/2>

• If u mod 4 = 3, then

note(t) = |(u-11)/4 (-a+b+c)/2 (a-b+c)/2 1+(a+b-c)/2>

If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad, and <4 6 9 11|note(n, t) = n.

The keenanismic tablet

This is based on the five keenanismic tetrads, which are essentially tempered chords tempering out 385/384, plus the otonal and utonal tetrads. If we take the intervals for an otonal tetrad in close position, 5/4-6/5-7/6-8/7, we can permute them in 24 ways. The number of circular permutations, leading to chords which are not simply transpositions, is six. Keenanismic tempering makes all six chords dyadic. in addition the steps 8/7-5/4-8/7-14/11 lead to a keenanismic tempered 35/32-5/4-3/2-12/7 chord.

Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos |* w x y>. If the chord is transposed by some quantity with monzo M, the chord product is changed to |* x y z> + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions:

1-5/4-3/2-7/4 => [1 1 1]

1-8/7-4/3-8/5 => [3 3 3]

1-5/4-3/2-12/7 => [0 1 3]

1-6/5-3/2-7/4 => [0 3 1]

35/32-5/4-3/2-7/4 => [1 0 0]

1-5/4-35/24-7/4 => [3 0 0]

35/32-5/4-3/2-12/7 => [0 0 2]

Here 35/32-5/4-3/2-12/7 is the 8/7-6/5-8/7-14/11 chord. Also, in place of 1-8/7-4/3-8/5 we can use 1-6/5-3/2-12/7 => [-1 3 3].

If we have a 3-tuple c representing a keenanismic tetrad and reduce it modulo four, define k(c) by reversing the above correspondence, except that we assign k([3 3 3]) = [1,6/5,3/2,12/7]. If w is a keenanismic tetrad, define q(w) by taking the product w[1]w[2]w[3]w[4], finding the monzo m, and returning [2-m[2] 2-m[3] 2-m[4]]. Define kord(c) by finding t = (q(k(c)) - c)/4 and setting kord(c) = 3^t[1] 5^t[2] 7^t[3] k(c). Now we may define note(n, c) by setting u = kord(c), v = <4 6 9 11|u[1], i = v mod 4, and then setting note(n, c) = 2^((n-v-i)/4) u[i+1]. We then have our basic identity <4 6 9 11|note(n, c) = n.

Given any keenanismic or otonal/utonal tetrad, and counting octave equivalent tetrads as the same, any tetrad will share a common triad with either four or five other tetrads. If we make tetrads the verticies of a graph where the edges join two tetrads if and only if they share a common triad, we obtain an infinite, but locally finite, connected graph. It is possible to get from one tetrad to any other solely by means of chord relationships where there is a common triad.

If a and b are two allowable triples of integers representing tetrads (so that their reduction modulo four is one of the seven allowed types) then we may derive a positive definite quaratic form on 3-tuples [x y z] by Q(x, y, z) = 4x^2 + 3y^2 + 3z^2 + 4xy + 4xz + 2xy, such that Q(a-b) = 8 if a and b are 3-tuples representing tetrads with a common triad. The square root √Q(a-b) is a Eulidean measure of distance, and √8 is the minimum distance between tetrad 3-tuples. Such 3-tuple pairs a minimum distance apart share at least an interval in common, and usually a triad, and are therefore closely related.

The pele tablet

This is a tablet for the rank 3 13-limit temperament pele. It is based on the following 71 chords, in the 5-limit transversal:

chords = [

[1, 6/5, 3/2, 9/5],

[1, 5/4, 3/2, 1280/729],

[1, 6/5, 3/2, 2187/1280],

[1, 6561/5120, 3/2, 9/5],

[1, 2560/2187, 3/2, 5/3],

[1, 2560/2187, 3/2, 1280/729],

[1, 5/4, 3/2, 327680/177147],

[1, 6/5, 3/2, 531441/327680],

[1, 6561/5120, 3/2, 531441/327680],

[1, 2560/2187, 3/2, 327680/177147],

[1, 9/8, 81920/59049, 1280/729],

[1, 81/64, 10240/6561, 1280/729],

[1, 5/4, 81920/59049, 1280/729],

[1, 5/4, 729/512, 59049/32768],

[1, 4782969/4194304, 177147/131072, 5/3],

[1, 655360/531441, 20971520/14348907, 5/3],

[1, 9/8, 81920/59049, 2621440/1594323],

[1, 32/27, 20971520/14348907, 2621440/1594323],

[1, 4782969/4194304, 3/2, 5/3],

[1, 2097152/1594323, 3/2, 9/5],

[1, 10/9, 5/4, 1594323/1048576],

[1, 5/4, 81920/59049, 2621440/1594323],

[1, 32/27, 2097152/1594323, 2621440/1594323],

[1, 6561/5120, 531441/327680, 9/5],

[1, 9/8, 5/4, 2621440/1594323],

[1, 5/4, 729/512, 1280/729],

[1, 9/8, 729/512, 128/81],

[1, 655360/531441, 81920/59049, 1280/729]]

If c is a chord identifier, [c[1] c[2] c[3]], c[2] and c[3] any integers and 0 < c[1] < 72, then if u = n - 6c[2] - 9c[3], v = u mod 4 and w = chords(c[1]), we may define note(n, c) = 2^((u-v)/4) 3^c[2] 5^c[3] w[v+1]. The steps of the 71 chords are as follows:

• 5-limit JI
• Chord 1

6/5-5/4-6/5-10/9

• 7-limit JI
• Chords 2-6

5/4-6/5-7/6-8/7

6/5-5/4-8/7-7/6

9/7-7/6-6/5-10/9

7/6-9/7-10/9-6/5

7/6-9/7-7/6-8/7

• 11-limit JI
• Chords 7-15

5/4-6/5-11/9-12/11

6/5-5/4-12/11-11/9

9/7-7/6-12/11-11/9

7/6-9/7-11/9-12/11

9/8-11/9-14/11-8/7

14/11-11/9-9/8-8/7

5/4-11/10-14/11-8/7

5/4-8/7-14/11-11/10

11/10-14/11-9/7-10/9

• 13-limit JI
• Chords 16-26

15/13-13/11-11/9-6/5

11/9-13/11-15/13-6/5

9/8-11/9-13/11-16/13

13/11-11/9-9/8-16/13

15/13-13/10-10/9-6/5

13/10-15/13-6/5-10/9

10/9-9/8-16/13-13/10

5/4-11/10-13/11-16/13

13/11-11/10-5/4-16/13

9/7-14/11-11/10-10/9

9/8-10/9-13/10-16/13

• 441/440 tempered
• Chords 27-34

5/4-8/7-11/9-8/7

9/8-14/11-11/10-14/11

11/9-9/8-14/11-8/7

9/8-11/9-8/7-14/11

9/8-8/7-11/9-14/11

9/8-14/11-11/9-8/7

8/7-11/9-9/7-10/9

9/7-11/9-8/7-10/9

• 896/891 tempered
• Chords 35-37

9/8-11/9-9/8-9/7

9/8-11/9-9/7-9/8

11/9-9/8-9/8-9/7

• 196/195 tempered
• Chord 38

7/6-6/5-7/6-16/13

• 352/351 tempered
• Chords 39-53

9/8-13/10-5/4-12/11

11/9-16/13-9/8-13/11

15/13-13/10-9/8-13/11

9/8-10/9-16/13-13/10

5/4-13/10-9/8-12/11

13/10-9/8-16/13-10/9

11/9-16/13-10/9-6/5

9/8-13/10-10/9-16/13

11/9-16/13-11/9-12/11

16/13-11/9-9/8-13/11

16/13-11/9-13/11-9/8

16/13-11/9-6/5-10/9

11/9-16/13-13/11-9/8

13/10-15/13-13/11-9/8

10/9-9/8-13/10-16/13

• 364/363 tempered
• Chords 54-62

13/11-14/11-8/7-7/6

13/11-14/11-11/9-12/11

11/9-13/11-12/11-14/11

13/11-14/11-12/11-11/9

12/11-13/11-11/9-14/11

14/11-13/11-12/11-11/9

14/11-13/11-7/6-8/7

13/11-7/6-13/11-16/13

14/11-13/11-11/9-12/11

• {196/195, 352/351} tempered
• Chords 63-64

11/9-16/13-7/6-8/7

16/13-11/9-8/7-7/6

• {196/195, 363/363} tempered
• Chords 65-66

7/6-13/11-13/11-16/13

13/11-13/11-7/6-16/13

• {352/351, 364/363] tempered
• Chords 67-69

7/6-9/7-9/8-13/11

14/11-13/11-9/8-13/11

9/7-7/6-13/11-9/8

• Pele {196/195, 352/351, 364/363} tempered
• Chords 70-71

13/11-14/11-9/8-13/11

14/11-13/11-13/11-9/8]]:

5et tablets

The 7-limit 5et tablet

If we define u = n - 9a - 5b - 3c then supposing a+b+c is even,

• If u mod 5 = 0, then

note(t) = |u/5 (-a+b+c)/2 (a-b+c)/2 (a+b-c)/2>

• If u mod 5 = 1, then

note(t) = |(u-16)/5 2+(-a+b+c)/2 (a-b+c)/2 (a+b-c)/2>

• If u mod 5 = 2, then

note(t) = |(u-12)/5 (-a+b+c)/2 1+(a-b+c)/2 (a+b-c)/2>

• If u mod 5 = 3, then

note(t) = |(u-8)/5 1+(-a+b+c)/2 (a-b+c)/2 (a+b-c)/2>

• If u mod 5 = 4, then

note(t) = |(u-14)/5 (-a+b+c)/2 (a-b+c)/2 1+(a+b-c)/2>

Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit pentad, where <5 8 12 14|note(n, t) = n.

The meantone add6/9 tablet is based on the meantone add6/9 pentad, which can also be called the add2/9 pentad, the meantone pentatonic scale or Meantone[5]. The tablet is extremely simple, consiting of an ordered pair [n, c], where we have a meantone transversal for the notes defined by u = n-8c, where

• If u mod 5 = 0 then

note(n, c) = |u/5 c>

• If u mod 5 = 1 then

note(n, c) = |(u-1)/5-3 c+2>

• If u mod 5 = 2 then

note(n, c) = |(u-2)/5-6 c+4>

• If u mod 5 = 3 then

note(n, c) = |(u-3)/5-1 c+1>

• If u mod 5 = 4 then

note(n, c) = |(u-4)/5-4 c+3>

In all cases <5 8|note(n, c) = n. Tempering the Pythgorean transversal by flattening 3 gives, as usual, a meantone tuning.

The 5et portent tablet

This is based on the following twelve chords, which are expressed in terms of the 2.5.7 transversal of the 11-limit rank three temperament portent, which tempers out 385/384, 441/440 and hence also 1029/1024 and 3025/3024.

chords = [[1, 131072/117649, 5/4, 512/343, 7/4], [1, 131072/117649, 1048576/823543, 512/343, 1048576/588245], [1, 131072/117649, 16384/12005, 512/343, 7/4], [1, 131072/117649, 1048576/823543, 512/343, 80/49], [1, 2048/1715, 16384/12005, 512/343, 7/4], [1, 35/32, 5/4, 512/343, 4096/2401], [1, 35/32, 5/4, 12005/8192, 7/4], [1, 35/32, 5/4, 512/343, 7/4], [1, 131072/117649, 5/4, 10/7,7/4], [1, 588245/524288, 5/4, 10/7, 7/4], [1, 131072/117649, 16384/12005, 131072/84035, 7/4], [16384/16807, 131072/117649, 5/4, 10/7, 7/4]]

If now we set a chord identifier c = [c[1] c[2] c[3]], where c[1] ranges from 1 to 12, picking out the corresponding chord in the chords list. The other two values, c[2] and c[3], transpose the root of the chords by 5^c[2] 7^c[3]. If u = n - 12c[2] - 14c[3], v = u mod 5, and w = chords(c[1]), then

note(n, [c[1] c[2] c[3]]) = 2^((u-v)/5) 5^c[2] 7^c[3] w[v+1].

Once again, <5 8 12 14 17|note(n, c) = c.

The selection of these particular representatives for each of the twelve types of chords is based on each of them having a common triad—three common notes—in common with the utonal pentad, the first chord in the chords list. The chord representatives do not need to be given in terms of a 2.5.7 subgroup transversal, and a more perspicuous way of expressing the same portent temperament pentads is

1-9/8-5/4-3/2-7/4

1-9/8-9/7-3/2-9/5

1-9/8-11/8-3/2-7/4

1-9/8-9/7-3/2-18/11

• Keenanismic (385/384 essentially tempered) pentads

1-6/5-11/8-3/2-7/4

1-12/11-5/4-3/2-12/7

1-12/11-5/4-16/11-7/4

1-12/11-5/4-3/2-7/4

• Werckismic (441/440 essentially tempered) pentads

1-9/8-5/4-10/7-7/4

1-10/9-5/4-10/7-7/4

1-9/8-11/8-11/7-7/4

55/56-9/8-5/4-10/7-7/4

Each of the other eleven chords shares a triad with the first, otonal, pentad, and all of the chords can be related by an infinite but locally finite graph by drawing an edge between chords with a common triad.

The 6et tutone tutonic tablet

This tablet is based on the tutonic sextad, which in terms of the 99/98 (Huygens) version of 11-limit meantone consists of a chain of five tones, followed by an augmented second; in other words a {81/80, 126/125, 99/98}-tempered version of 9/8-9/8-9/8-9/8-9/8-8/7, which in terms of notes rather than steps is a tempered 1-9/8-5/4-7/5-11/7-7/4. Using this chord as the basis for harmony puts one in tutone temperament, a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale.

If the tablet is the ordered pair [n, c] and if u = n-19c, then if i = u mod 6, define note(n, c) = |(u-i)/6-3i 2c+2i>. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity <12 19|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as <6 19|note(n, c) = n, where note(n, c) in subgroup monzo terms is |(u-i)/6-3i c+i>.

The 13-limit 7et tablet

Let <r e3 e5 e7 e11 e13| denote an otonal 13-limit septad with root given by |* e3 e5 e7 e11 e13> when r is even, which in close position is 9/8-5/4-11/8-3/2-13/8-7/4-2. If r is odd, let it denote a utonal pentad which in close position is 12/11-6/5-4/3-3/2-12/7-24/13-2

We will regard the 6-tuple [r e3 e5 e7 e11 e13] as denoting a septad, and [n, [r e3 e5 e7 e11 e13]] the corresponding tablet. We first calculate u = n - 11e3 - 16e5 - 20e7 - 24e11 - 26e13. Then if r is even, we have

• If u mod 7 = 0, then

note(n, [r e3 e5 e7 e11 e13]) = |u/7 e3 e5 e7 e11 e13>

• If u mod 7 = 1, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-22)/7 e3+2 e5 e7 e11 e13>

• If u mod 7 = 2, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-16)/7 e3 e5+1 e7 e11 e13>

• If u mod 7 = 3, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-24)/7 e3 e5 e7 e11+1 e13>

• If u mod 7 = 4, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-11)/7 e3+1 e5 e7 e11 e13>

• If u mod 7 = 5, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-26)/7 e3 e5 e7 e11 e13+1>

• If u mod 7 = 6, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-20)/7 e3 e5 e7+1 e11 e13>

On the other hand, if r is odd:

• If u mod 7 = 0, then

note(n, [r e3 e5 e7 e11 e13]) = |u/7 e3 e5 e7 e11 e13>

• If u mod 7 = 1, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-1)/7+2 e3+1 e5 e7 e11-1 e13>

• If u mod 7 = 2, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-2)/7+1 e3+1 e5-1 e7 e11 e13>

• If u mod 7 = 3, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-3)/7+2 e3-1 e5 e7 e11+1 e13>

• If u mod 7 = 4, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-4)/7-1 e3+1 e5 e7 e11 e13>

• If u mod 7 = 5, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-5)/7+2 e3+1 e5 e7-1 e11 e13+1>

• If u mod 7 = 6, then

note(n, [r e3 e5 e7 e11 e13]) = |(u-6)/7+3 e3+1 e5 e7 e11 e13-1>

The tablet satisfies the identity

<7 11 16 20 24 26|note(n, [r e3 e5 e7 e11 e13]) = n.